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In general, minimal energy maps do not exist \\cite{BeMi1}. When $\\dom$ has a single hole, Berlyand and Rybalko \\cite{BeRy1} proved that for small $\\v$ local minimizers do exist. We extend the result in \\cite{BeRy1}: $\\d E_\\v(u)$ has, in domains $\\dom$ with $2,3,...$ holes and for small $\\v$, local minimizers. Our approach is very similar to the one"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1111.1571","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-07T13:31:41Z","cross_cats_sorted":[],"title_canon_sha256":"0dfe3eb43eee7487bb89c6c87e0d2fde14abdcf23a4484bd0406264675a94796","abstract_canon_sha256":"fb3ce0de555c61b19ef2cdcb2c67c536af3ea3e017754b3f37d1566bc7b04329"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:25.775346Z","signature_b64":"Ywg1UY6YeQaXNdAquuG7qyjlkrjmbsqluk1z1UjBBtGIqzl9r1FAgxA5QkTsi1hDYbFaUE1h6qtrfYSfcr0zDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"86360575ce7d41b71334c5793cf3b6c1a921ab6099a103a81be8f80ec77d042e","last_reissued_at":"2026-05-18T04:09:25.774600Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:25.774600Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Local Minimizers of the Ginzburg-Landau Functional with Prescribed Degrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Micka\\\"el Dos Santos (ICJ)","submitted_at":"2011-11-07T13:31:41Z","abstract_excerpt":"We consider, in a smooth bounded multiply connected domain $\\dom\\subset\\R^2$, the Ginzburg-Landau energy $\\d E_\\v(u)=1/2\\int_\\dom{|\\n u|^2}+\\frac{1}{4\\v^2}\\int_\\dom{(1-|u|^2)^2}$ subject to prescribed degree conditions on each component of $\\p\\dom$. 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